Number Patterns and Sequences | 数字模式与数列

📚 Number Patterns and Sequences | 数字模式与数列

In mathematics, patterns are everywhere. A number pattern is a list of numbers that follows a specific rule, and a sequence is the ordered list of numbers formed by that rule. In this lesson, we will learn how to identify rules, extend sequences, find missing terms, and describe patterns using precise mathematical language.

在数学中，模式无处不在。数字模式是按照特定规则排列的一列数字，而数列则是依据该规则形成的有序数字列表。在本课中，我们将学习如何识别规则、延伸数列、寻找缺失项，并用准确的数学语言描述模式。

1. What Is a Number Pattern? | 什么是数字模式？

A number pattern is a set of numbers arranged in a special order. Each number in the pattern is called a term. To find the next term, we must first discover the rule that connects one term to the next.

数字模式是按特定顺序排列的一组数字。模式中的每个数字称为一项。要找到下一项，我们必须先发现连接相邻两项的规则。

For example, look at this pattern: 2, 4, 6, 8, 10, … The rule is “add 2 each time.” So the next term would be 12.

例如，观察这个模式：2, 4, 6, 8, 10, … 规则是”每次加2″。因此下一项是12。

Patterns can be created with addition, subtraction, multiplication, division, or even a combination of operations. Some patterns grow quickly, while others grow slowly.

模式可以通过加法、减法、乘法、除法甚至多种运算组合来创建。有些模式增长很快，有些则增长缓慢。

2. Arithmetic Sequences | 等差序列

An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference. It can be positive, negative, or zero.

等差序列是指任意两个连续项之间的差都相等的数列。这个恒定的差称为公差。公差可以是正数、负数或零。

Consider the sequence: 5, 8, 11, 14, 17, … Each time, we add 3. So the common difference is +3. To find the next term, simply add 3 to 17, giving 20.

考虑数列：5, 8, 11, 14, 17, … 每次加3。因此公差是+3。要求下一项，只需将17加3，得到20。

What about a decreasing sequence? Look at 20, 17, 14, 11, … Here the common difference is -3. The next term is 11 – 3 = 8.

那递减数列呢？观察20, 17, 14, 11, … 这里公差是-3。下一项是11 – 3 = 8。

General form: a, a + d, a + 2d, a + 3d, …

通项形式：a, a + d, a + 2d, a + 3d, …

Here, a is the first term and d is the common difference. The n-th term can be found using the formula:

这里，a是首项，d是公差。第n项可以通过以下公式求出：

n-th term = a + (n – 1) × d

第n项 = a + (n – 1) × d

For the sequence 5, 8, 11, 14, …, the 10th term is 5 + (10 – 1) × 3 = 5 + 27 = 32.

对于数列5, 8, 11, 14, …，第10项是5 + (10 – 1) × 3 = 5 + 27 = 32。

3. Geometric Sequences | 等比序列

A geometric sequence is a sequence where each term is multiplied by a constant to get the next term. This constant is called the common ratio. The ratio can be greater than 1, between 0 and 1, or even negative.

等比序列是每一项都乘以一个常数得到下一项的数列。这个常数称为公比。公比可以大于1、介于0和1之间，甚至可以是负数。

For example, 3, 6, 12, 24, 48, … Here, each term is multiplied by 2. The common ratio is 2. The next term is 48 × 2 = 96.

例如，3, 6, 12, 24, 48, … 这里每一项都乘以2。公比是2。下一项是48 × 2 = 96。

What about 100, 50, 25, 12.5, …? Here each term is multiplied by 0.5 (or divided by 2). The next term is 12.5 × 0.5 = 6.25.

那100, 50, 25, 12.5, …呢？这里每一项都乘以0.5（或除以2）。下一项是12.5 × 0.5 = 6.25。

General form: a, a × r, a × r², a × r³, …

通项形式：a, a × r, a × r², a × r³, …

The n-th term of a geometric sequence is given by:

等比数列的第n项公式为：

n-th term = a × r ⁽ⁿ⁻¹⁾

第n项 = a × r ⁽ⁿ⁻¹⁾

For the sequence 3, 6, 12, 24, …, the 6th term is 3 × 2⁵ = 3 × 32 = 96.

对于数列3, 6, 12, 24, …，第6项是3 × 2⁵ = 3 × 32 = 96。

4. Triangular Numbers | 三角形数

Triangular numbers are formed by counting the number of dots needed to make an equilateral triangle. The first few triangular numbers are 1, 3, 6, 10, 15, 21, …

三角形数是通过计算排列成等边三角形所需的点数而形成的。前几个三角形数是1, 3, 6, 10, 15, 21, …

The pattern works like this: add 2 to get from 1 to 3, add 3 to get from 3 to 6, add 4 to get from 6 to 10, and so on. The increments increase by 1 each time.

这个模式是这样的：从1到3加2，从3到6加3，从6到10加4，依此类推。每次增加的量都递增1。

The n-th triangular number can be calculated using:

第n个三角形数可以用以下公式计算：

Tₙ = n × (n + 1) ÷ 2

For example, the 7th triangular number is 7 × 8 ÷ 2 = 28. Check: 21 + 7 = 28. Correct!

例如，第7个三角形数是7 × 8 ÷ 2 = 28。验证：21 + 7 = 28。正确！

Triangular numbers appear in many real-life situations, such as arranging bowling pins or counting handshakes in a group.

三角形数出现在许多实际情境中，比如排列保龄球瓶或计算一组人中的握手次数。

5. Square and Cube Numbers | 平方数与立方数

Square numbers are formed by multiplying a whole number by itself. The sequence of square numbers is 1, 4, 9, 16, 25, 36, …

平方数是一个整数与自身相乘得到的。平方数序列是1, 4, 9, 16, 25, 36, …

Notice that the differences between consecutive square numbers increase: 3, 5, 7, 9, 11, … These differences are consecutive odd numbers.

注意相邻平方数之间的差在增大：3, 5, 7, 9, 11, … 这些差是连续的奇数。

The n-th square number is simply n². For example, the 8th square number is 8² = 64.

第n个平方数就是n²。例如，第8个平方数是8² = 64。

Cube numbers are formed by multiplying a whole number by itself twice. The sequence is 1, 8, 27, 64, 125, … The n-th cube number is n³.

立方数是一个整数与自身相乘两次得到的。序列是1, 8, 27, 64, 125, … 第n个立方数是n³。

Here is a comparison table:

以下是一个对照表：

n	Square n²	Cube n³
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125

Square and cube numbers are very useful in geometry when calculating areas and volumes.

平方数和立方数在几何中计算面积和体积时非常有用。

6. Fibonacci Sequence | 斐波那契数列

The Fibonacci sequence is one of the most famous patterns in mathematics. It begins with 0 and 1, and each subsequent term is the sum of the two previous terms.

斐波那契数列是数学中最著名的模式之一。它以0和1开始，之后每一项都是前两项之和。

The sequence starts: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

数列以：0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … 开始。

Let us check: 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on.

让我们验证：0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8，依此类推。

The Fibonacci sequence appears in nature: in the arrangement of leaves, the spirals of sunflowers, pinecones, and even in the branching of trees.

斐波那契数列出现在自然界中：叶子排列、向日葵的螺旋、松果，甚至树木的分枝方式中。

Each term is the sum of the previous two terms.

每一项都是前两项之和。
The rule can be written as: Fₙ = Fₙ₋₁ + Fₙ₋₂

该规则可以写为：Fₙ = Fₙ₋₁ + Fₙ₋₂

If the 8th term is 21 and the 9th term is 34, then the 10th term is 21 + 34 = 55.

如果第8项是21，第9项是34，那么第10项是21 + 34 = 55。

7. Finding Missing Terms | 寻找缺失项

Sometimes a sequence is given with one or more missing terms. To find the missing term, first identify the rule, then apply it to the surrounding terms.

有时一个数列中会有一个或多个缺失项。要找到缺失项，首先要识别规则，然后将其应用于相邻的项。

Example 1: Find the missing term in 4, 9, 14, ?, 24.

例1：找出4, 9, 14, ?, 24中的缺失项。

The difference between consecutive known terms is 5. So the missing term is 14 + 5 = 19. Check: 19 + 5 = 24. Correct.

已知相邻项之间的差是5。因此缺失项是14 + 5 = 19。验证：19 + 5 = 24。正确。

Example 2: Find the missing term in 2, 6, 18, ?, 162.

例2：找出2, 6, 18, ?, 162中的缺失项。

Each term is multiplied by 3. So the missing term is 18 × 3 = 54. Check: 54 × 3 = 162. Correct.

每一项都乘以3。因此缺失项是18 × 3 = 54。验证：54 × 3 = 162。正确。

Example 3: Find the missing term in 1, 4, ?, 16, 25.

例3：找出1, 4, ?, 16, 25中的缺失项。

These are square numbers: 1², 2², 3², 4², 5². The missing term is 3² = 9.

这些是平方数：1², 2², 3², 4², 5²。缺失项是3² = 9。

8. Special Number Patterns | 特殊数字模式

Beyond arithmetic and geometric sequences, there are many other interesting number patterns. Let us explore a few.

除了等差和等比序列之外，还有许多其他有趣的数字模式。让我们探索几种。

The powers of 2: 2, 4, 8, 16, 32, 64, 128, … Each term is doubled.

2的幂：2, 4, 8, 16, 32, 64, 128, … 每一项都翻倍。

The powers of 10: 10, 100, 1000, 10000, … Each term is multiplied by 10.

10的幂：10, 100, 1000, 10000, … 每一项都乘以10。

Even numbers: 2, 4, 6, 8, 10, … and odd numbers: 1, 3, 5, 7, 9, … are simple patterns with a common difference of 2.

偶数：2, 4, 6, 8, 10, … 和奇数：1, 3, 5, 7, 9, … 是公差为2的简单模式。

There are also patterns like 1, 11, 121, 1331, 14641, … which are related to Pascal’s triangle. Notice that 121 = 11², 1331 = 11³.

还有像1, 11, 121, 1331, 14641, … 这样的模式，它们与帕斯卡三角有关。注意121 = 11²，1331 = 11³。

When identifying a pattern, always look for the relationship between consecutive terms. Ask yourself:

识别模式时，始终寻找相邻项之间的关系。问问自己：

Is a constant number being added or subtracted?

是否在加或减一个常数？
Is a constant number being multiplied or divided?

是否在乘或除以一个常数？
Are the operations changing? If so, how?

运算是否在变化？如果是，怎么变化？

9. Real-World Applications | 实际应用

Number patterns are not just abstract ideas — they help us solve real problems every day.

数字模式不仅仅是抽象的概念——它们帮助我们每天解决实际问题。

When saving money, if you save $5 in the first week, $10 in the second week, $15 in the third week, your savings form an arithmetic sequence. You can predict your total savings after 10 weeks.

存钱时，如果第一周存5美元，第二周存10美元，第三周存15美元，你的存款就构成了一个等差序列。你可以预测10周后的总存款。

In construction, bricklayers use patterns to stack bricks. In computer science, geometric sequences describe how data grows. In biology, Fibonacci numbers describe how populations of rabbits can grow under ideal conditions.

在建筑中，砌砖工人用模式来堆叠砖块。在计算机科学中，等比序列描述数据如何增长。在生物学中，斐波那契数描述兔群在理想条件下如何增长。

Situation: A school gym has rows of seats. The first row has 20 seats, and each row has 4 more seats than the row before it. How many seats are in the 6th row?

情境：学校体育馆有一排排座位。第一排有20个座位，每一排比前一排多4个座位。第6排有多少个座位？

This is an arithmetic sequence: 20, 24, 28, 32, 36, 40. The 6th row has 40 seats.

这是一个等差序列：20, 24, 28, 32, 36, 40。第6排有40个座位。

10. Practice Problems | 练习题

Try solving these problems on your own before checking the answers.

先自己尝试解决这些问题，再核对答案。

Problem 1: Find the next three terms of the sequence 7, 11, 15, 19, …

题1：找出数列7, 11, 15, 19, …的接下来三项。

Problem 2: What is the 12th term of the arithmetic sequence 3, 7, 11, 15, …?

题2：等差序列3, 7, 11, 15, …的第12项是多少？

Problem 3: Find the missing term in the geometric sequence 5, 15, 45, ?, 405.

题3：找出等比序列5, 15, 45, ?, 405中的缺失项。

Problem 4: Is 64 a square number, a cube number, or both?

题4：64是平方数、立方数，还是两者都是？

Problem 5: The 6th term of a Fibonacci-like sequence is 21 and the 7th term is 34. What is the 8th term?

题5：在类斐波那契数列中，第6项是21，第7项是34。第8项是多少？

Answers:

答案：

Answer 1: The common difference is 4. The next three terms are 23, 27, 31.

答案1：公差是4。接下来三项是23, 27, 31。

Answer 2: Use the formula a + (n – 1) × d = 3 + 11 × 4 = 3 + 44 = 47.

答案2：使用公式a + (n – 1) × d = 3 + 11 × 4 = 3 + 44 = 47。

Answer 3: The common ratio is 3. The missing term is 45 × 3 = 135.

答案3：公比是3。缺失项是45 × 3 = 135。

Answer 4: 64 = 8² and 64 = 4³, so 64 is both a square number and a cube number.

答案4：64 = 8²，且64 = 4³，所以64既是平方数也是立方数。

Answer 5: The 8th term is 21 + 34 = 55.

答案5：第8项是21 + 34 = 55。

11. Summary | 总结

Number patterns and sequences help us organize information and make predictions. We learned about arithmetic sequences with a constant difference, geometric sequences with a constant ratio, triangular numbers, square and cube numbers, and the famous Fibonacci sequence.

数字模式与数列帮助我们组织信息并做出预测。我们学习了具有恒定公差的等差序列、具有恒定公比的等比序列、三角形数、平方数与立方数，以及著名的斐波那契数列。

Key formulas to remember:

需要记住的关键公式：

Arithmetic n-th term: a + (n – 1) × d

等差第n项：a + (n – 1) × d
Geometric n-th term: a × r ⁽ⁿ⁻¹⁾

等比第n项：a × r ⁽ⁿ⁻¹⁾
Triangular number: n × (n + 1) ÷ 2

三角形数：n × (n + 1) ÷ 2
Square number: n²

平方数：n²
Cube number: n³

立方数：n³

Always identify the rule first, then extend the sequence confidently.

始终先识别规则，然后自信地延伸数列。

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Number Patterns and Sequences | 数字模式与数列